Gelfand Kirillov Conjecture Research Articles

Poisson Noether's Problem and Poisson rationality

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces.

Some Open Problems in the Context of Skew PBW Extensions and Semi-graded Rings

In this paper, we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre’s conjecture will be discussed for a particular class of skew PBW extensions. The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings. Advances for the solution of some of the problems are included.

An extension of [formula omitted] related to the alternating group and Galois orders

In 2010, V. Futorny and S. Ovsienko gave a realization of U(gln) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of S1×S2×⋯×Sn, where Sj is the symmetric group on j variables. An interesting question is what a similar algebra would be in the invariant ring with respect to a product of alternating groups. In this paper we define such an algebra, denoted A(gln), and show that it is a Galois ring. For n=2, we show that it is a generalized Weyl algebra, and for n=3 provide generators and a list of verified relations. We also discuss some techniques to construct Galois orders from Galois rings. Additionally, we study categories of finite-dimensional modules and generic Gelfand-Tsetlin modules over A(gln). Finally, we discuss connections between the Gelfand-Kirillov Conjecture, A(gln), and the positive solution to Noether's problem for the alternating group.

Algebras of invariant differential operators

We prove that an invariant subalgebra AnW of the Weyl algebra An is a Galois order over an adequate commutative subalgebra Γ when W is a two-parameters irreducible unitary reflection group G(m,1,n), m≥1, n≥1, including the Weyl group of type Bn, or alternating group An, or the product of n copies of a cyclic group of fixed finite order. Earlier this was established for the symmetric group in [15]. In each of the cases above, except for the alternating groups, we show that AnW is free as a right (left) Γ-module. Similar results are established for the algebra of W-invariant differential operators on the n-dimensional torus where W is a symmetric group Sn or orthogonal group of type Bn or Dn. As an application of our technique we prove the quantum Gelfand-Kirillov conjecture for Uq(sl2), the first Witten deformation and the Woronowicz deformation.

Quantum linear Galois orders

We define a class of quantum linear Galois algebras which include the universal enveloping algebra the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand–Zetlin algebras of type A, the subalgebras of G-invariants of the quantum affine space, quantum torus for and of the quantum Weyl algebra for G = Sn. We show that all quantum linear Galois algebras satisfy the quantum Gelfand-Kirillov conjecture. Moreover, it is shown that the subalgebras of invariants of the quantum affine space and of quantum torus for the reflection groups and of the quantum Weyl algebra for symmetric groups are, in fact, Galois orders over an adequate commutative subalgebras and free as right (left) modules over these subalgebras. In the rank 1 cases the results hold for an arbitrary finite group of automorphisms when the field is

Noncommutative Noether’s problem for complex reflection groups

We solve the noncommutative Noether's problem for the reflection groups by showing that the skew field of the invariants of the Weyl algebra under the action of any reection group is a Weyl field, that is isomorphic to a skew field of some Weyl algebra over a transcendental extension of the ground field. We also extend this result to the invariants of the ring of differential operators on any dimensional torus.The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.

The q-difference Noether problem for complex reflection groups and quantum OGZ algebras

ABSTRACTFor any complex reflection group G = G(m,p,n), we prove that the G-invariants of the division ring of fractions of the n:th tensor power of the quantum plane is a quantum Weyl field and give explicit parameters for this quantum Weyl field. This shows that the q-difference Noether problem has a positive solution for such groups, generalizing previous work by Futorny and the author [10]. Moreover, the new result is simultaneously a q-deformation of the classical commutative case and of the Weyl algebra case recently obtained by Eshmatov et al. [8].Second, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk [18] originating in the classical Gelfand–Tsetlin formulas. Special cases of quantum OGZ algebras include the quantized enveloping algebra of 𝔤𝔩n and quantized Heisenberg algebras. We show that any quantum OGZ algebra can be naturally realized as a Galois ring in the sense of Futorny-Ovsienko [11], with symmetry group being a direct product of complex reflection groups G(m,p,rk).Finally, using these results, we prove that the quantum OGZ algebras satisfy the quantum Gelfand–Kirillov conjecture by explicitly computing their division ring of fractions.

On the Gelfand–Kirillov conjecture for the W-algebras attached to the minimal nilpotent orbits

Consider the W-algebra H attached to the minimal nilpotent orbit in a simple Lie algebra g over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand–Kirillov conjecture holds for such a W-algebra, then it holds for the universal enveloping algebra U(g). This, together with a result of A. Premet, implies that the analogue of the Gelfand–Kirillov conjecture fails for some W-algebras attached to the minimal nilpotent orbit in Lie algebras of types Bn(n≥3), Dn(n≥4), E6,E7,E8, and F4.

Some aspects of noncommutative invariant theory and the Noether’s problem

This survey discusses the classical Noether’s problem and presents some important cases where it has a positive solution. With the Weyl algebras taking place of the polynomial algebras, some aspects of noncommutative invariant theory are introduced: in particular, the noncommutative version of the Noether’s problem. Some known cases are discussed and some new results are presented. As an application one obtains a new proof of the Gelfand–Kirillov conjecture for $$gl_n$$ . There is a striking similarity between the cases with positive solution of the Noether’s problem for both commutative and noncommutative versions, and this leads to many conjectures.

ORE AND GOLDIE THEOREMS FOR SKEW PBW EXTENSIONS

Many rings and algebras arising in quantum mechanics can be interpreted as skew Poincaré–Birkhoff–Witt (PBW) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this paper, we extend the classical Ore and Goldie theorems, known for skew polynomial rings, to this wide class of non-commutative rings. As application, we prove the quantum version of the Gelfand–Kirillov conjecture for the skew quantum polynomials.

Solution of a $$q$$ q -difference Noether problem and the quantum Gelfand–Kirillov conjecture for $$\mathfrak gl _N$$ g l N

It is shown that the $$q$$ -difference Noether problem for all classical Weyl groups has a positive solution, simultaneously generalizing well known results on multisymmetric functions of Mattuck (Proc Am Math Soc 19:764–765, 1968) and Miyata (Nagoya Math J 41:69–73, 1971) in the case $$q=1$$ , and $$q$$ -deforming the noncommutative Noether problem for the symmetric group (Futorny et al. in Adv Math 223:773–796, 2010). It is also shown that the quantum Gelfand–Kirillov conjecture for $$\mathfrak gl _N$$ (for a generic $$q$$ ) follows from the positive solution of the $$q$$ -difference Noether problem for the Weyl group of type $$D_n$$ . The proof is based on the theory of Galois rings (Futorny and Ovsienko in J Algebra 324:598–630, 2010). From here we obtain a proof of the quantum Gelfand–Kirillov conjecture for $$\mathfrak gl _N$$ , and for a certain extension of $$\mathfrak sl _N$$ . Previously, the case of $$\mathfrak sl _N$$ was shown by Fauquant-Millet (J Algebra 218:93–116, 1999) and by Alev and Dumas (J Algebra 170:229–265, 1994) (for $$N=2,3$$ ). Moreover, we give an explicit description of the skew fields of fractions for $$U_q(\mathfrak gl _N)$$ and $$U_q^\mathrm{ext}(\mathfrak sl _N)$$ which generalizes the results of Alev and Dumas (J Algebra 170:229–265, 1994).

Enveloping algebras of Slodowy slices and Goldie rank

Let \( U\left( {\mathfrak{g},e} \right) \) be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra \( \mathfrak{g} = {\text{Lie}}(G) \) and let I be a primitive ideal of the enveloping algebra \( U\left( \mathfrak{g} \right) \) whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that \( I = {\text{An}}{{\text{n}}_{U\left( \mathfrak{g} \right)}}\left( {{Q_e}{ \otimes_{U\left( {\mathfrak{g},e} \right)}}V} \right) \) for some finite dimensional irreducible \( U\left( {\mathfrak{g},e} \right) \)-module V, where Q e stands for the generalised Gelfand–Graev \( \mathfrak{g} \)-module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient \( {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} \) always divides dim V. For \( \mathfrak{g} = \mathfrak{s}{\mathfrak{l}_n} \), we use a theorem of Joseph on Goldie fields of primitive quotients of \( U\left( \mathfrak{g} \right) \) to establish the equality \( {\text{rk}}\left( {{{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}} \right) = \dim V \). We show that this equality continues to hold for \( \mathfrak{g} \ncong \mathfrak{s}{\mathfrak{l}_n} \) provided that the Goldie field of \( {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} \) is isomorphic to a Weyl skew-field and use this result to disprove Joseph’s version of the Gelfand–Kirillov conjecture formulated in the mid-1970s.

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p > 0 . Let Z be the centre of the universal enveloping algebra U = U ( g ) of g . Its maximal spectrum is called the Zassenhaus variety of g . We show that, under certain mild assumptions on G, the field of fractions Frac ( Z ) of Z is G-equivariantly isomorphic to the function field of the dual space g ∗ with twisted G-action. In particular Frac ( Z ) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac ( Z ) , a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand–Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G 2 .

The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant's Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand–Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand–Tsetlin modules using Gelfand–Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand–Tsetlin modules for finite W-algebras.

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand–Kirillov conjecture, Invent. Math. 56 (3) (1980) 191–213] in many cases. We also present a general setup of decategorification, precategorification and categorification.

Gelfand–Kirillov conjecture in positive characteristics

Let g be a finite-dimensional Lie algebra over an algebraically closed field of characteristic p > 0 and let U ( g ) be the universal enveloping algebra of g . We show in this paper that the division ring of fractions of U ( g ) is isomorphic to the ring of fractions of a Weyl algebra in the following cases: for g = gl n or sl n if p ∤ n , for the Witt algebra W 1 and for some tensor product W 1 ⊗ A of W 1 with a truncated polynomial ring. Furthermore we also show that the centre of U ( g ) in the last two cases is a unique factorisation domain, in accordance with recent results of Premet, Tange, Braun and Hajarnavis.

SUR LES INVARIANTS DE Bλ SOUS L'ACTION DE SOUS-GROUPES FINIS D'AUTOMORPHISMES: CONJECTURE DE GELFAND-KIRILLOV ET HOMOLOGIE DE HOCHSCHILD

Let B λ be a minimal primitive quotient of U(sl 2) and G a finite subgroup of automorphisms of B λ. We prove that the algebras of invariants BG λ, satisfy the Gelfand-Kirillov conjecture, that is Frac (BG λ) ≃ D 1(C) (where D 1(C) denotes the 1st Weyl field). Then we compute the 0th Hochschild homology group of these algebras.

Fields of Fractions of Quantum Solvable Algebras

We introduce the notion of pure Q-solvable algebra. The quantum matrices, quantum Weyl algebra, Uq(n) are the examples. It is proved that the skew field of fractions of a pure Q-solvable algebra R is isomorphic to the skew field of twisted rational functions. This is a quantum version of the Gelfand–Kirillov conjecture for solvable algebraic Lie algebras.

The Gelfand–Kirillov Conjecture for Lie Algebras of Dimension at Most Eight

Univ Reims, Dept Math, F-51687 Reims 2, France. Univ Limburg, Dept Math, B-3590 Diepenbeek, Belgium.Alev, J, Univ Reims, Dept Math, BP 1039, F-51687 Reims 2, France.